Solve for $n$, $ \dfrac{8}{4n + 4} = -\dfrac{3}{5n + 5} + \dfrac{n - 10}{n + 1} $
Answer: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4n + 4$ $5n + 5$ and $n + 1$ The common denominator is $20n + 20$ To get $20n + 20$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{8}{4n + 4} \times \dfrac{5}{5} = \dfrac{40}{20n + 20} $ To get $20n + 20$ in the denominator of the second term, multiply it by $\frac{4}{4}$ $ -\dfrac{3}{5n + 5} \times \dfrac{4}{4} = -\dfrac{12}{20n + 20} $ To get $20n + 20$ in the denominator of the third term, multiply it by $\frac{20}{20}$ $ \dfrac{n - 10}{n + 1} \times \dfrac{20}{20} = \dfrac{20n - 200}{20n + 20} $ This give us: $ \dfrac{40}{20n + 20} = -\dfrac{12}{20n + 20} + \dfrac{20n - 200}{20n + 20} $ If we multiply both sides of the equation by $20n + 20$ , we get: $ 40 = -12 + 20n - 200$ $ 40 = 20n - 212$ $ 252 = 20n $ $ n = \dfrac{63}{5}$